986.433 Rhombic Dodecahedron: The next and last of the hierarchy
of primitive
allspace-fillers is the rhombic dodecahedron. The rhombic
dodecahedron is the domain of
a sphere (see Sec.
981.13). The rhombic dodecahedron
consists of 12 Kites__ergo, 48 A +
mods, 48 A - mods, 24 B + mods, and 24 B - mods. Sum-total
number of
modules...144

986.434
This is the limit set of simplest allspace-fillers
associable within one nuclear
domain of closest-packed spheres and their respective
interstitial spaces. There are other
allspace-fillers that occur in time-size multiplications
of nuclear domains, as for instance
the tetrakaidecahedron. (Compare Sec.
950.12.)

986.440
Table: Set of Simple Allspace-fillers
This completes one spheric domain (i.e., sphere plus
interstitial space) of one unit-radius
sphere in closest packing, each sphere being centered
at every other vertex of the isotropic
vector matrix.

Name:

FaceTriangles

Type Hedra

A QuantaModules

B QuantaModules

Sum- TotalModules

MITE

4

tetrahedron

2

1

3

SYTE

BITE

4

tetrahedron

4

2

6

RITE

4

tetrahedron

4

2

6

LITE

6

hexahedron

4

2

6

KITE

KATE

5

pentahedron

8

4

12

KAT

5

pentahedron

8

4

12

OCTET

6

hexahedron

12

6

18

COUPLER

8

octahedron

16

8

24

CUBE

6

hexahedron

48

24

72

RHOMBICDODECAHEDRON

12

dodecahedron

96

48

144

(For the minimum time-size special case realizations
of the two-frequency systems. multiply each of the
above Quanta Module numbers by eight.)

986.450 Energy Aspects of Spherical Modular Arrays

986.451
The rhombic dodecahedron has an allspace-filling function
as the domain of
any one sphere in an aggregate of unit-radius, closest-packed
spheres; its 12 mid-diamond-
face points C are the points of intertangency of all
unit-radius, closest-packed sphere
aggregates; wherefore that point C is the midpoint of
every vector of the isotropic vector
matrix, whose every vertex is the center of one of the
unit-radius, closest-packed spheres.

986.452
These 12 inter-closest-packed-sphere-tangency points__the
C points__are
the 12 exclusive contacts of the "Grand Central Station"
through which must pass all the
great-circle railway tracks of most economically interdistanced
travel of energy around any
one nuclear center, and therefrom__through the C points__to
other spheres in Universe.
These C points of the rhombic dodecahedron's mid-diamond
faces are also the energetic
centers-of-volume of the Couplers, within which there
are 56 possible unique
interarrangements of the A and B Quanta Modules.

986.453
We next discover that two ABABO pentahedra of any two
tangentially
adjacent, closest-packed rhombic dodecahedra will produce
an asymmetric octahedron
OABABO' with O and O' being the volumetric centers (nuclear
centers) of any two
tangentially adjacent, closest-packed, unit-radius spheres.
We call this nucleus-to-nucleus,
asymmetric octahedron the Coupler, and we found that
the volume of the Coupler is
exactly equal to the volume of one regular tetrahedron__i.e.,
24 A Quanta Modules. We
also note that the Coupler always consists of eight
asymmetric and identical tetrahedral
Mites, the minimum simplex allspace-filling of Universe,
which Mites are also identifiable
with the quarks (Sec.
1052.360).

986.454
We then discover that the Mite, with its two energy-conserving
A Quanta
Modules and its one energy-dispersing B Quanta Module
(for a total combined volume of
three quanta modules), serves as the cosmic minimum
allspace-filler, corresponding
elegantly (in all ways) with the minimum-limit case
behaviors of the nuclear physics'
quarks. The quarks are the smallest discovered "particles";
they always occur in groups of
three, two of which hold their energy and one of which
disperses energy. This quite clearly
identifies the quarks with the quanta module of which
all the synergetics hierarchy of
nuclear concentric symmetric polyhedra are co-occurrent.

986.455
In both the rhombic triacontahedron of tetravolume
5 and the rhombic
dodecahedron of tetravolume 6 the distance from system
center O at AO is always greater
than CO, and BO is always greater than AO.

986.456
With this information we could reasonably hypothesize
that the
triacontahedron of tetravolume 5 is that static polyhedral
progenitor of the only-
dynamically-realizable sphere of tetravolume 5, the
radius of which (see Fig.
986.314) is
only 0.04 of unity greater in length than is the prime
vector radius OC, which governs the
dimensioning of the triacontahedron's 30 midface cases
of 12 right-angled corner
junctions around mid-diamond-vertex C, which provides
the 12 right angles around C-the
four right-angled corners of the T Quanta Module's ABC
faces of their 120 radially
arrayed tetrahedra, each of which T Quanta Module has
a volume identical to that of the A
and B Quanta Modules.

986.457
We also note that the radius OC is the same unitary
prime vector with which
the isotropic vector matrix is constructed, and it is
also the VE unit-vector-radius distance
outwardly from O, which O is always the common system
center of all the members of the
entire cosmic hierarchy of omniconcentric, symmetric,
primitive polyhedra. In the case of
the rhombic triacontahedron the 20 OA lines' distances
outwardly from O are greater than
OC, and the 12 OB lines' distances are even greater
in length outwardly from O than OA.
Wherefore I realized that, when dynamically spun, the
greatcircle chord lines AB and CB
are centrifugally transformed into arcs and thus sprung
apart at B, which is the outermost
vertex__ergo, most swiftly and forcefully outwardly impelled.
This centrifugal spinning
introduces the spherical excess of 6 degrees at the
spherical system vertex B. (See Fig.
986.405)
Such yielding increases the spheric appearance of
the spun triacontahedron, as
seen in contradistinction to the diamond-faceted, static,
planar-bound, polyhedral state
aspect.

986.458
The corners of the spherical triacontahedron's 120
spherical arc-cornered
triangles are 36 degrees, 60 degrees and 90 degrees,
having been sprung apart from their
planar-phase, chorded corners of 31.71747441 degrees,
58.28252559 degrees, and 90
degrees, respectively. Both the triacontahedron's chorded
and arced triangles are in
notable proximity to the well-known 30-, 60-, and 90-degree-cornered
draftsman's flat,
planar triangle. I realized that it could be that the
three sets of three differently-distanced-
outwardly vertexes might average their outward-distance
appearances at a radius of only
four percent greater distance from O-thus producing
a moving-picture-illusioned
"dynamic" sphere of tetravolume 5, having very mildly
greater radius than its static,
timeless, equilibrious, rhombic triacontahedron state
of tetravolume 5 with unit-vector-
radius integrity terminaled at vertex C.

986.459
In the case of the spherical triacontahedron the total
spherical excess of
exactly 6 degrees, which is one-sixtieth of unity =
360 degrees, is all lodged in one corner.
In the planar case 1.71747441 degrees have been added
to 30 degrees at corner B and
subtracted from 60 degrees at corner A. In both the
spherical and planar triangles__as well
as in the draftsman's triangle__the 90-degree corners
remain unchanged.

986.460
The 120 T Quanta Modules radiantly arrayed around the
center of volume of
the rhombic triacontahedron manifest the most spherical
appearance of all the hierarchy of
symmetric polyhedra as defined by any one of the seven
axially rotated, great circle system
polyhedra of the seven primitive types of great-circle
symmetries.

986.461
What is the significance of the spherical excess of
exactly 6 degrees? In the
transformation from the spherical rhombic triacontahedron
to the planar triacontahedron
each of the 120 triangles releases 6 degrees. 6 × 120
= 720. 720 degrees = the sum of the
structural angles of one tetrahedron = 1 quantum of
energy. The difference between a
high-frequency polyhedron and its spherical counterpart
is always 720 degrees, which is
one unit of quantum__ergo, it is evidenced that spinning
a polyhedron into its spherical
state captures one quantum of energy__and releases it
when subsiding into its pre-time-
size primitive polyhedral state.

986.471
A series of considerations leads to the definition
of the most spherical-
appearing limit of triangular subdivisioning:

recalling that the experimentally demonstrable "most
spherically-appearing"
structure is always in primitive reality a polyhedron;

recalling that the higher the modular frequency
of a system the more spheric it
appears, though it is always polyhedral and approaching
not a "true sphere"
limit but an unlimited multiplication of its polyhedral
facetings;

recalling that the 120 outer surface triangles of
the icosahedron's 15 great
circles constitute the cosmic maximum limit of system-surface
omni-triangular-
self-subdivisioning into centrally collected tetrahedron
components; and

recalling that the icosahedron's 10- and 6-great-circle
equators of spin further
subdivide the 15 great circles' outer 120 LCD triangles
into four different right
triangles, ADC, CDE, CFE, and EFB (see Fig.
901.03),

then it becomes evident that the icosahedron's three
sets of symmetrical greatcircle
spinnabilities__i.e., 6 + 10 + 15 (which totals 31 great
circle self-halvings)__generate a
total of 242 unit-radius, external vertexes, 480 external
triangles, and 720 internal triangles
(which may be considered as two congruent internal triangles,
each being one of the
internal triangular faces of the 480 tetrahedra whose
480 external triangular faces are
showing-in which case there are 1440 internal triangles).
The 480 tetrahedra consist of
120 OCAD, 120 OCDE, 120 OCEF, and 120 OFEB tetrahedra.
(See Fig.
986.471.) The
480 internal face-congruent tetrahedra therefore constitute
the "most spheric-appearing"
of all the hemispheric equators' self-spun, surface-subdividing
entirely into triangles of all
the great circles of all the primitive hierarchy of
symmetric polyhedra.

986.472
In case one thinks that the four symmetrical sets of
the great circles of the
spherical VE (which total 25 great circles in all) might
omnisubdivide the system surface
exclusively into a greater number of triangles, we note
that some of the subdivision areas
of the 25 great circles are not triangles (see quadrant
BCEF in Fig.
453.01__third printing
of Synergetics 1__of which quadrangles there are a total
of 48 in the system); and note
that the total number of triangles in the 25-great-circle
system is 288__ergo, far less than
the 31 great circles' 480 spherical right triangles;
ergo, we become satisfied that the
icosahedron's set of 480 is indeed the cosmic maximum-limit
case of system-self-spun
subdivisioning of its self into tetrahedra, which 480
consist of four sets of 120 similar
tetrahedra each.

986.473
It then became evident (as structurally demonstrated
in reality by my
mathematically close-toleranced geodesic domes) that
the spherical trigonometry
calculations' multifrequenced modular subdividing of
only one of the icosahedron's 120
spherical right triangles would suffice to provide all
the basic trigonometric data for any
one and all of the unit-radius vertex locations and
their uniform interspacings and
interangulations for any and all frequencies of modular
subdividings of the most
symmetrical and most economically chorded systems' structuring
of Universe, the only
variable of which is the special case, time-sized radius
of the special-case system being
considered.

986.474
This surmise regarding nature's most-economical, least-effort
design
strategy has been further verified by nature's own use
of the same geodesics mathematics
as that which I discovered and employed in my domes.
Nature has been using these
mathematical principles for eternity. Humans were unaware
of that fact. I discovered these
design strategies only as heretofore related, as an
inadvertent by-product of my
deliberately undertaking to find nature's coordination
system. That nature was manifesting
icosahedral and VE coordinate patterning was only discovered
by other scientists after I
had found and demonstrated geodesic structuring, which
employed the synergetics'
coordinate-system strategies. This discovery by others
that my discovery of geodesic
mathematics was also the coordinate system being manifest
by nature occurred after I had
built hundreds of geodesic structures around the world
and their pictures were widely
published. Scientists studying X-ray diffraction patterns
of protein shells of viruses in 1959
found that those shells disclosed the same patterns
as those of my widely publicized
geodesic domes. When Dr. Aaron Klug of the University
of London__who was the one
who made this discovery__communicated with me, I was
able to send him the
mathematical formulae for describing them. Klug explained
to me that my geodesic
structures are being used by nature in providing the
"spherical" enclosures of her own
most critical design-controlling programming devices
for realizing all the unique
biochemical structurings of all biology__which device
is the DNA helix.

986.475
The structuring of biochemistry is epitomized in the
structuring of the
protein shells of all the viruses. They are indeed all
icosahedral geodesic structures. They
embracingly guard all the DNA-RNA codified programming
of all the angle-and-frequency
designing of all the biological, life-accommodating,
life-articulating structures. We find
nature employing synergetics geometry, and in particular
the high-frequency geodesic
"spheres," in many marine organisms such as the radiolaria
and diatoms, and in structuring
such vital organs as the male testes, the human brain,
and the eyeball. All of these are
among many manifests of nature's employment on her most
critically strategic occasions
of the most cosmically economical, structurally effective
and efficient enclosures, which
we find are always mathematically based on multifrequency
and three-way-triangular
gridding of the "spherical"__because high-frequenced__icosahedron,
octahedron, or
tetrahedron.

986.476
Comparing the icosahedron, octahedron, and tetrahedron-the
icosahedron
gives the most volume per unit weight of material investment
in its structuring; the high-
frequency tetrahedron gives the greatest strength per
unit weight of material invested; and
the octahedron affords a happy__but not as stable-mix
of the two extremes, for the
octahedron consists of the prime number 2, 22 = 4; whereas
the tetrahedron is the odd
prime number 1 and the icosahedron is the odd prime
number 5. Gear trains of even
number reciprocate, whereas gear trains of an odd number
of gears always lock; ergo, the
tetrahedral and icosahedral geodesic systems lock-fasten
all their structural systems, and
the octahedron's compromise, middle-position structuring
tends to yield transformingly
toward either the tetra or the icosa locked-limit capabilities__either
of which tendencies is
pulsatively propagative.